Abstract
In the modern theory of partial differential equations each point value or integral value is a functional and the Riesz elements of linear functionals are the Green’s functions so that a critical examination of the accuracy of FE-results must concentrate on the role Green’s functions play in the FE-context, theoretically and practically. Most functionals are unbounded and so the FE-method, strictly speaking, transgresses the bounds of the theory but it does so very successfully. The analogy between Green’s functions and Lagrange multipliers finally provides estimates for nonlinear problems at the linearization point.
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- 1.
For not to confuse the vector space with the shear force \(V\) we write \(\mathcal{ V} \) and \(\mathcal{ V} _h\) as well.
- 2.
The so called taxi norm—distance by grid lines (blocks)—and the Euclidean norm—as the raven flies—is also a pair of equivalent norms.
- 3.
In higher dimensions these terms would be genuine boundary integrals.
- 4.
The indices on the Green’s functions and the Dirac deltas are to distinguish the different functions, see Sect. 2.4.1.
- 5.
You see a gap only in 1-D. In 2-D you can only sense it if you circle the gap once. Then you will experience a displacement shift in the direction of the dislocation.
- 6.
We restrain at this point from introducing quadrupoles or even octupoles [5].
- 7.
Actually the complete Green’s function is \(1/(4\,\pi \,r)\). But in the literature the potential is always given in the form (2.258), that is the \(1/4\,\pi \) must be contained in the constant \(G\).
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Hartmann, F. (2013). Basic Concepts. In: Green's Functions and Finite Elements. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29523-2_2
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DOI: https://doi.org/10.1007/978-3-642-29523-2_2
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